Bayesian Predictive Distributions Research Articles

Bayesian typhoon precipitation prediction with a mixture of ensemble forecast-based and historical event-based prediction functions

Statistical post-processing has been widely adopted in probabilistic forecasting, especially when a predictive distribution of weather variables is of interest. Among the available post-processing methods, a common tool is the ensemble Bayesian model averaging. Challenges arise when it is applied to heavy rain prediction, including the precipitation produced by tropical cyclones. The issues are related to the rare extreme event and limited available data for model training. To address this, we propose in this research a Bayesian mixture model, Bmix, based on information from forecast members and historical tropical cyclones, rather than solely on immediately observed precipitation data. This approach provides a grid-specific Bayesian predictive distribution and an easy-to-derive probabilistic forecast with posterior samples generated from the Markov chain Monte Carlo algorithm. This post-processing procedure can be executed on each grid simultaneously, reducing the computational time significantly. In addition, a categorized Bmix (C.Bmix) is implemented to accommodate different scenarios such as heavy or torrential rain in applications. The proposed approaches are demonstrated on the 24-hour accumulated precipitation forecast of a category 4-equivalent super typhoon Dujuan landing in Taiwan in 2015; while two others, Matmo and Soudelor, are adopted to provide historical information. Over the Taiwan main island, a total of 1282 grids are considered, each with a resolution of 5 km and 20 ensemble member forecasts from the Weather Research and Forecasting Ensemble Prediction System (WEPS) at a 6-hour interval during the typhoon period. The analyses show that the proposed mixture approaches present larger percentages of positive CRPSS values than traditional BMA. For instance, at the time when Dujuan made landfall, the percentages are 92.2 % for Bmix, 95.2 % for C.Bmix, and 15.4 % for BMA when the true precipitation amount exceeds 200 mm/24 h. The proposed methods outperform in the probability prediction and the overall pattern of precipitation.

Statistical Inference of Normal Distribution Based on Several Divergence Measures: A Comparative Study

Statistical predictive analysis is a very useful tool for predicting future observations. Previous literature has addressed both Bayesian and non-Bayesian predictive distributions of future statistics based on past sufficient statistics. This study focused on evaluating Bayesian and Wald predictive-density functions of a future statistic V based on a past sufficient statistic W obtained from a normal distribution. Several divergence measures were used to assess the closeness of the predictive densities to the future density. The difference between these divergence measures was investigated, using a simulation study. A comparison between the two predictive densities was examined, based on the power of a test. The application of a real data set was used to illustrate the results in this article.

Martingale posterior distributions

Abstract The prior distribution is the usual starting point for Bayesian uncertainty. In this paper, we present a different perspective that focuses on missing observations as the source of statistical uncertainty, with the parameter of interest being known precisely given the entire population. We argue that the foundation of Bayesian inference is to assign a distribution on missing observations conditional on what has been observed. In the i.i.d. setting with an observed sample of size n, the Bayesian would thus assign a predictive distribution on the missing Yn+1:∞ conditional on Y1:n, which then induces a distribution on the parameter. We utilize Doob’s theorem, which relies on martingales, to show that choosing the Bayesian predictive distribution returns the conventional posterior as the distribution of the parameter. Taking this as our cue, we relax the predictive machine, avoiding the need for the predictive to be derived solely from the usual prior to posterior to predictive density formula. We introduce the martingale posterior distribution, which returns Bayesian uncertainty on any statistic via the direct specification of the joint predictive. To that end, we introduce new predictive methodologies for multivariate density estimation, regression and classification that build upon recent work on bivariate copulas.

The Cost of Imperfect Knowledge: How Epistemic Uncertainties Influence Flood Hazard Assessments

AbstractClassical approaches to flood hazard are obtained by the concatenation of a recurrence model for the events (i.e., an extreme river discharge) and an inundation model that propagates the discharge into a flood extent. The classical approach, however, uses “best‐fit” models that do not include uncertainty from incomplete knowledge or limited data availability. The inclusion of these, so called epistemic uncertainties, can significantly impact flood hazard estimates and the corresponding decision‐making process. We propose a simulation approach to robustly account for uncertainty in model's parameters, while developing a useful probabilistic output of flood hazard for further risk assessments via the Bayesian predictive posterior distribution of water depths. A Peaks‐Over‐Threshold Bayesian analysis is performed for future events simulation, and a pseudo‐likelihood probabilistic approach for the calibration of the inundation model is used to compute uncertain water depths. The annual probability averaged over all possible models’ parameters is used to develop hazard maps that account for epistemic uncertainties. Results are compared to traditional hazard maps, showing that not including epistemic uncertainties can underestimate the hazard and lead to non‐conservative designs, and that this trend increases with return period. Results also show that the influence of the uncertainty in the future occurrence of discharge events is predominant over the inundation simulator uncertainties for the case study.

Bayesian predictive distributions of oil returns using mixed data sampling volatility models

This study explores the benefits of incorporating fat-tailed innovations, asymmetric volatility response, and an extended information set into crude oil return modeling and forecasting. To this end, we utilize standard volatility models such as Generalized Autoregressive Conditional Heteroskedastic (GARCH), Generalized Autoregressive Score (GAS), and Stochastic Volatility (SV), along with Mixed Data Sampling (MIDAS) regressions, which enable us to incorporate the impacts of relevant financial/macroeconomic news into asset price movements. For inference and prediction, we employ an innovative Bayesian estimation approach called the density-tempered sequential Monte Carlo method. Our findings indicate that the inclusion of exogenous variables is beneficial for GARCH-type models while offering only a marginal improvement for GAS and SV-type models. Notably, GAS-family models exhibit superior performance in terms of in-sample fit, out-of-sample forecast accuracy, as well as Value-at-Risk and Expected Shortfall prediction.

Testing the forecasting skills of aftershock models using a Bayesian framework

The Epidemic Type Aftershock Sequence (ETAS) model and the modified Omori law (MOL) are two aftershock rate models that are used for operational earthquake/aftershock forecasting. Previous studies have investigated the relative performance of the two models for specific case studies. However, a rigorous comparative evaluation of the forecasting performance of the basic aftershock rate models for several different earthquake sequences has not been done before. In this study, forecasts of five prominent aftershock sequences from multiple catalogs are computed using the Bayesian predictive distribution, which fully accounts for the uncertainties in the model parameters. This is done by the Markov Chain Monte Carlo (MCMC) sampling of the model parameters and forward simulation of the ETAS or MOL models to compute the aftershock forecasts. The forecasting results are evaluated using five different statistical tests, including two comparison tests. The forecasting skill tests indicate that the ETAS model tends to perform consistently well on the first three tests. The MOL fails the same tests for certain forecasting time intervals. However, in the comparison tests, it is not definite whether the ETAS model is the better performing model. This work demonstrates the use of forecast testing for different catalogs, which is also applicable to catalogs with a higher magnitude of completeness.

Bayesian Predictive Distributions for Imbalance Prices With Time-Varying Factor Impacts

A dynamic Bayesian model is developed to estimate the time-varying nature of the drivers of the system imbalance prices in the British electricity market. We find that the key exogenous factors that significantly influence prices have impacts that evolve substantially over time. Thus, by modeling their evolution with time varying parameter estimation and making conditional forecasts on the latest estimates, more accurate forecasts are produced. Furthermore, using a Bayesian approach allows predictive distributions to be developed, as would be required for value-at-risk compliance purposes. These densities are also found to be more accurate at the extreme quantiles than a conventional GARCH model with static parameters. We validated the superior performance of this Bayesian time varying predictive density method with the same data as in a previously published benchmark model.

Agent-based model projections for reducing HIV infection among MSM: Prevention and care pathways to end the HIV epidemic in Chicago, Illinois.

Our objective is to improve local decision-making for strategies to end the HIV epidemic using the newly developed Levers of HIV agent-based model (ABM). Agent-based models use computer simulations that incorporate heterogeneity in individual behaviors and interactions, allow emergence of systemic behaviors, and extrapolate into the future. The Levers of HIV model (LHM) uses Chicago neighborhood demographics, data on sex-risk behaviors and sexual networks, and data on the prevention and care cascades, to model local dynamics. It models the impact of changes in local preexposure prophylaxis (PrEP) and antiretroviral treatment (ART) (ie, levers) for meeting Illinois' goal of "Getting to Zero" (GTZ) -reducing by 90% new HIV infections among men who have sex with men (MSM) by 2030. We simulate a 15-year period (2016-2030) for 2304 distinct scenarios based on 6 levers related to HIV treatment and prevention: (1) linkage to PrEP for those testing negative, (2) linkage to ART for those living with HIV, (3) adherence to PrEP, (4) viral suppression by means of ART, (5) PrEP retention, and (6) ART retention. Using tree-based methods, we identify the best scenarios at achieving a 90% HIV infection reduction by 2030. The optimal scenario consisted of the highest levels of ART retention and PrEP adherence, next to highest levels of PrEP retention, and moderate levels of PrEP linkage, achieved 90% reduction by 2030 in 58% of simulations. We used Bayesian posterior predictive distributions based on our simulated results to determine the likelihood of attaining 90% HIV infection reduction using the most recent Chicago Department of Public Health surveillance data and found that projections of the current rate of decline (2016-2019) would not achieve the 90% (p = 0.0006) reduction target for 2030. Our results suggest that increases are needed at all steps of the PrEP cascade, combined with increases in retention in HIV care, to approach 90% reduction in new HIV diagnoses by 2030. These findings show how simulation modeling with local data can guide policy makers to identify and invest in efficient care models to achieve long-term local goals of ending the HIV epidemic.

Enriched standard conjugate priors and the right invariant prior for Wishart distributions

The prediction of the variance–covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback–Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these prior distributions. Furthermore, we studied the performance of the Bayesian predictive distribution based on the reference prior distribution in the family and showed that there exists a prior distribution in the family that dominates the reference prior distribution. Our study provides new insight into the multivariate analysis when there exists an ordered inferential importance for the independent variables.

Incorporating Measurement Error in Astronomical Object Classification

Most general-purpose classification methods, such as support-vector machine (SVM) and random forest (RF), fail to account for an unusual characteristic of astronomical data: known measurement error uncertainties. In astronomical data, this information is often given in the data but discarded because popular machine learning classifiers cannot incorporate it. We propose a simulation-based approach that incorporates heteroscedastic measurement error into an existing classification method to better quantify uncertainty in classification. The proposed method first simulates perturbed realizations of the data from a Bayesian posterior predictive distribution of a Gaussian measurement error model. Then, a chosen classifier is fit to each simulation. The variation across the simulations naturally reflects the uncertainty propagated from the measurement errors in both labeled and unlabeled data sets. We demonstrate the use of this approach via two numerical studies. The first is a thorough simulation study applying the proposed procedure to SVM and RF, which are well-known hard and soft classifiers, respectively. The second study is a realistic classification problem of identifying high-z (2.9 ≤ z ≤ 5.1) quasar candidates from photometric data. The data are from merged catalogs of the Sloan Digital Sky Survey, the Spitzer IRAC Equatorial Survey, and the Spitzer-HETDEX Exploratory Large-Area Survey. The proposed approach reveals that out of 11,847 high-z quasar candidates identified by a random forest without incorporating measurement error, 3146 are potential misclassifications with measurement error. Additionally, out of 1.85 million objects not identified as high-z quasars without measurement error, 936 can be considered new candidates with measurement error.

Fourier-Analysis-Based Form of Normalized Maximum Likelihood: Exact Formula and Relation to Complex Bayesian Prior

Normalized maximum likelihood (NML) distribution of probabilistic model gives the optimal code length function in the sense of minimax regret. Despite its optimal property, the calculation of NML distribution is not easy, and existing efficient methods have been focusing on its asymptotic behavior, or on specific models. This paper gives an efficient way to calculate NML by integral on parameter domain, not on data domain, showing that NML distribution is a Bayesian predictive distribution with a complex prior, based on our novel Fourier expansion approach. Our results provide an integrated way to calculate NML for exponential family and also include a non-asymptotic version of previous work on asymptotic behavior for general cases. The applications of our methodology are not limited to but also include normal distribution, Gamma distribution, Weibull distribution, and von Mises distribution.

1215Considerations for using multiple imputation in propensity score-weighted analysis

1215Considerations for using multiple imputation in propensity score-weighted analysis

Information criteria and cross validation for Bayesian inference in regular and singular cases

In data science, an unknown information source is estimated by a predictive distribution defined from a statistical model and a prior. In an older Bayesian framework, it was explained that the Bayesian predictive distribution should be the best on the assumption that a statistical model is convinced to be correct and a prior is given by a subjective belief in a small world. However, such a restricted treatment of Bayesian inference cannot be applied to highly complicated statistical models and learning machines in a large world. In 1980, a new scientific paradigm of Bayesian inference was proposed by Akaike, in which both a model and a prior are candidate systems and they had better be designed by mathematical procedures so that the predictive distribution is the better approximation of unknown information source. Nowadays, Akaike’s proposal is widely accepted in statistics, data science, and machine learning. In this paper, in order to establish a mathematical foundation for developing a measure of a statistical model and a prior, we show the relation among the generalization loss, the information criteria, and the cross-validation loss, then compare them from three different points of view. First, their performances are compared in singular problems where the posterior distribution is far from any normal distribution. Second, they are studied in the case when a leverage sample point is contained in data. And last, their stochastic properties are clarified when they are used for the prior optimization problem. The mathematical and experimental comparison shows the equivalence and the difference among them, which we expect useful in practical applications.

A comparison of estimators for the network autocorrelation model based on observed social networks

Network autocorrelation models (NAMs) are widely used to study a response variable of interest among subjects embedded within a network. Although the NAM is highly useful for studying such networked observational units, several simulation studies have raised concerns about point estimation. Specifically, these studies have consistently demonstrated a negative bias of maximum likelihood estimators (MLEs) of the network effect parameter. However, in order to gain a practical understanding of point estimation in the NAM, these findings need to be expanded in three important ways. First, these simulation studies are based on relatively simple network generative models rather than observed networks, thereby leaving as an open question how realistic network topologies may affect point estimation in practice. Second, although there has been strong work done in developing two-stage least squares estimators as well as Bayesian estimators, only the MLE has received extensive attention in the literature, thus leaving practitioners in question as to best practices. Third, the performance of these estimators need to be compared using both bias and variance, as well as the coverage rate of each estimator's corresponding confidence or credible interval. In this paper we describe a simulation study which aims to overcome these shortcomings in the following way. We first fit real social networks using the exponential random graph model and used the Bayesian predictive posterior distribution to generate networks with realistic topologies. We then compared the performance of the three different estimators mentioned above.

Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, for example the mean vector and the covariance matrix, are unknown and have to be estimated by using historical data on asset returns. Our new approach employs the Bayesian posterior predictive distribution which is the distribution of future realizations of asset returns given the observable sample. The parameters of posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. By contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step, and lead to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application not only gives the solution of the considered optimization problem, but also provides the posterior predictive distribution of the optimal portfolio return which can be used to construct a prediction interval. A Bayesian efficient frontier, the set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios which is known to be overoptimistic.

Fatigue life prediction of GFRP laminates using averaged Bayesian predictive distribution and Lamb wave velocity

This paper presents a fatigue life prediction approach for GFRP laminates using laser ultrasonic technique for data acquisition and a hierarchical Bayesian model for damage characterization as well as life prediction. Fatigue damage is measured as the degradation of the Lamb wave velocity and described using a scalar damage variable. Due to the complexity of the damage mechanisms involved in the fatigue process and the difficulties in distinguishing different damage modes, i.e. fiber breakage, matrix cracks and delamination, from a single damage variable, multiple degradation models of various levels of complexity are proposed to form a meta-model. Bayesian inference is applied to obtain the distributions of model parameters to include uncertainty information. Then a stacking approach is applied to average the predictive distributions to obtain the optimal combination of each sub-model to generate a meta-prediction with the least diverge from the experimental observation. Experimental results of both traditional stiffness data and Lamb wave velocity data are used for validation. A credible interval (CI) based criterion is also proposed for fatigue life prediction, where consistent results are achieved with reasonable use of informative priors.

A Full Bayesian Implementation of A Generalized Partial Credit Model with an Application to an International Disability Survey

Summary Generalized partial credit models (GPCMs) are ubiquitous in many applications in the health and medical sciences that use item response theory. Such polytomous item response models have a great many uses ranging from assessing and predicting an individual's latent trait to ordering the items to test the effectiveness of the test instrumentation. By implementing these models in a full Bayesian framework, computed through the use of Markov chain Monte Carlo methods implemented in the efficient STAN software package, the paper exploits the full inferential capability of GPCMs. The GPCMs include explanatory covariate effects which allow simultaneous estimation of regression and item parameters. The Bayesian methods for ranking the items by using the Fisher information criterion are implemented by using Markov chain Monte Carlo sampling. This allows us to propagate fully and to ascertain uncertainty in the inferences by calculating the posterior predictive distribution of the item-specific Fisher information criterion in a novel manner that has not been exploited in the literature before. Lastly, we propose a new Monte Carlo method for predicting the latent trait score of a new individual by approximating the relevant Bayesian predictive distribution. Data from a model disability survey carried out in Sri Lanka by the World Health Organization and the World Bank are used to illustrate the methods. The approaches proposed are shown to provide simultaneous model-based inference for all aspects of disability which can be explained by environmental and socio-economic factors.

Bayesian predictive distribution for a Poisson model with a parametric restriction

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.

Bayesian Predictive Distribution for a Negative Binomial Model

Estimation of the predictive probability function of a negative binomial distribution is addressed under the Kullback—Leibler risk. An identity that relates Bayesian predictive probability estimation to Bayesian point estimation is derived. Such identities are known in the cases of normal and Poisson distributions, and the paper extends the result to the negative binomial case. By using the derived identity, a dominance property of a Bayesian predictive probability is studied when the parameter space is restricted.

On Recursive Bayesian Predictive Distributions

ABSTRACTA Bayesian framework is attractive in the context of prediction, but a fast recursive update of the predictive distribution has apparently been out of reach, in part because Monte Carlo methods are generally used to compute the predictive. This article shows that online Bayesian prediction is possible by characterizing the Bayesian predictive update in terms of a bivariate copula, making it unnecessary to pass through the posterior to update the predictive. In standard models, the Bayesian predictive update corresponds to familiar choices of copula but, in nonparametric problems, the appropriate copula may not have a closed-form expression. In such cases, our new perspective suggests a fast recursive approximation to the predictive density, in the spirit of Newton’s predictive recursion algorithm, but without requiring evaluation of normalizing constants. Consistency of the new algorithm is shown, and numerical examples demonstrate its quality performance in finite-samples compared to fully Bayesian and kernel methods. Supplementary materials for this article are available online.